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Stress and Strain
This wiki page talks about stress and strain, their types and methodologies to measure them. Created by Faisal Ahmed Stress and Strain are two of the most important parameters in structural mechanics and therefore they play a paramount role in structural design of aircrafts. Aircrafts undergo different types of loading ( air loads, inertia loads, landing, taxi, powerplant,etc)[3]. These loads in turn create different types of stresses, which causes strains at different locations of an aircraft structure. Stress beyond permissible limit will cause failure of the aircraft structure. So, it is very important to do an accurate stress-strain analysis while designing an aircraft. Stress Definition Augustin-Louis Cauchy introduced the concept of stress in continuum mechanics around 1822. It is a measure of the average internal force exerted per unit area of a surface within a deformable body (Figure 1.1). There are two types of forces which cause stresses in a deformable body. One is surface force, acting on the surface of the body and the other one is body force, acting on each element of the body. For many practical applications body forces are negligible compared to the surface forces. So while defining stress, we use mainly surface forces [2]. ]] ]] Let us consider an arbitrary surface (external or internal) as shown in figure 1.1. A system of forces acts over a small area \Delta A of this surface in the vicinity of an arbitrary point P. Let, \Delta Fn be the resultant of this system of forces which does not necessarily coincide with the outer normal n''' associated with the element \Delta A . The average stress acting over the surface is obtained by dividing \Delta F_n with the infinitesimal area \Delta A . In the limit of \Delta A tending to zero, we obtain the quantity '''stress (Tn) acting on P as T_n = \lim_{\Delta A \to 0}\frac{\Delta F_n}{\Delta A} The line of action of resultant stress T_n and that of resultant force \Delta F_n coincide as illustrated in figure 1.2. The resultant stress T_n is a function of both the position of point P and the orientation of the surface. Types As shown in figure 1.2, the resultant stress T_n can be resolved in two components. Based on the orientation of the two components stress can be divided into two types, normal stress and shear stress [2]. Normal Stress ]] The component of T_n that is normal to the surface at point P (figure 1.2) is called normal stress. It is denoted by \sigma . For better understanding we can consider a column under distributed axial loading (figure 1.3). Normal stress is defined by \sigma = \frac{P}{A} Here P is the resultant of all the internal forces \Delta F and A is the summation of all the infinitesimal areas \Delta A . The above formula therefore represents average value of normal stress over the entire cross section, rather than the stress at a specific point on the surface. In reality the average value of stress differs from that at a particular point of the cross section, because the force distribution in the cross section is not uniform. But in practice we will assume a uniform force distribution in the cross section [1]. Shear Stress ]] The component of T_n that is tangential to the surface at point P (figure 1.2) is called shear stress. It is denoted by \tau . For better understanding we can consider a member under loading condition given as in figure 1.4. The transverse forces P and P' act on the member. An imaginary section is passes at point C between the points of application of the two forces. It is evident that internal forces will be generated at the plane of the section passing through C (figure 1.4). This internal force is called shear force. Then the average shearing stress is obtained by \tau_{ave} = \frac{P}{A} Here P is the resultant of the internal forces and A is the total cross sectional area. The value obtained from the formula above is an average value of the shearing stress over the entire section. Contrary to the case of normal stress the distribution of shearing stresses across a section cannot be assumed uniform [1]. ]] Units In SI system, unit for stress is Pascal (Pa), which is one Newton (force) per square meter (area). In engineering applications quantities are usually measured in megapascals (MPa) or gigapascals (GPa). In imperial unit system, stress is expressed in pounds-force per square inch (psi) or kilopounds-force per square inch (ksi). If a cubic body is acted upon by forces, the Cartesian components of stress acting upon the faces of this body (figure 1.5) is given by the tensor [1]. \mathbf{T} = \left( \begin{array}{ccc} \sigma_{x} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{y} & \tau_{yz}\\ \tau_{zx} & \tau_{zy} & \sigma_{z} \end{array} \right) Strain Definition When a body is subjected to system of forces, individual points of the body, in general will move. This movement is known as displacement, which is a vector quantity. If the displacement of different points of the body are different, each of them can be represented by its own vector. So, motion of a body experiencing a system of forces can be considered as a sum of two parts, (1) translation or rotation of the body as a whole, (2) movement of points within the body relative to each other. The first is known as the rigid body motion, which is applicable both the idealized rigid body and real deformable body. The second one known as deformation is a property of real bodies only [2]. Rigid body motion does not produce deformation. Strain'' is the geometrical measure of deformation of a body, representing the relative displacement between particles in that body. Types Strain and stress are two inseparable quantities. From the definition strain we can see that, stress is the cause and strain is the outcome. There will be no strain in a material, if there is no stress (i.e. no load) working on it. So, strains have the same classification as stresses, '''normal strain and shear strain [2]. Normal Strain ]] ]] Let us consider a rod BC of length L and uniform cross sectional area A as shown in figure 2.1. A normal load P is applied at the free end of the rod which creates a normal stress \sigma . We assume that the load is sufficient to create axial deformation in the body. Let, the change in length of the rod due to the load P is \delta . Then the normal strain is given by \epsilon = \frac{\delta}{L} As we assumed the rod to have uniform cross sectional area, normal stress \sigma can also be assumed to be uniform and it was appropriate to define strain \epsilon as the ratio of total deformation over the total length. In case of a varying cross section member (figure 2.2), strain should be defined at a particular point of that member [1]. In that case the strain is given by \epsilon = \lim_{\Delta x \to 0}\frac{\Delta \delta}{\Delta x} = \frac{d \delta}{dx} Normal stress and strain are related by Hooke's Law [1]., given by \sigma = E \epsilon . Here, E is called the modulus of elasticity. Shear Strain ]] ]] The shearing stresses as shown in figure 2.3 will tend to deform the cube into an oblique parallelepiped. Let us consider a unit cube (figure 2.3) subjected to no other stresses than shearing stresses \tau_{xy} and \tau_{yx} on faces of the cube perpendicular to the x an y axes. The cube will deform into a rhomboid of unit length (figure 2.4). Two of the angles formed at face xy will reduce from \frac{\pi}{2} to \frac{\pi}{2} - \gamma_{xy} and the other two will increase from \frac{\pi}{2} to \frac{\pi}{2} + \gamma_{xy} . The angle \gamma_{xy} is the shearing strain corresponding to x and y direction. If the deformation reduces the angle formed by the two faces oriented respectively toward positive x and y axes, \gamma_{xy} is positive, otherwise it is negative [1]. The relation between shear stress and shear strain is given by Hooke's Law [1]., expressed following \tau_{xy} = G \gamma_{xy} . Here G is called the modulus of rigidity. Units Normal strain \epsilon is a non-dimensional quantity so it has no units. Shear strain has the unit radian. Measurement Techniques There are various methods to measure stress and strain directly in structure undergoing deformation. Among the various methods some prominent ones that are used in aerospace industries are mentioned here. Strain Gage ]] Strain gages (figure 3.1) are used extensively wherever they are convenient to use. The strain gage technique is based on the fact that resistance of a wire increases with increasing strain and decreases with decreasing strain. Strain gages are made of semiconductors, can be very small in size and used particularly in research fields. Extensometers, which are large strain gages, are used to measure strains over 25 mm gage length [2]. Strain gages can give a measure of strain directly at any point of a structure under load. As stresses and strains are related by Hooke’s law, so stresses can be measured from strains and vice versa. Brittle Coating Brittle coating [4] is another method of measuring strain where the structure is coated with a special type of coating. Cracks in the coating appear due to loading of the part can be analyzed for direction and magnitude of the surface strains. Normally, the coating crack develops at right angles to directions of the maximum tensile strain. The coatings can be calibrated to obtain quantitative strain measurements. Brittle coating has some advantages. Its effective gage length approaches zero; it gives an overall view of the strain distribution and shows areas of stress concentrations; and it is applicable to any mechanical part of the structure, regardless of material, shape, or mode of loading. Brittle coatings are used to quickly locate and evaluate the high-stress points in a design and to obtain principal-stress directions for subsequent placement of electric-resistance strain gages. Optical Methods Optical methods [2] are other techniques of measuring stresses and strains. Photoelasticity, Moire Interferometry, Shadow Moire are among some of the optical methods used. ]] ]] Photoelasticity Photoelasticity is the easiest one to use among other optical methods [2]. It is based on the temporary double refraction behavior of transparent noncrystalline material which allows the material to behave like a crystal when it is stressed. A fringe pattern appears on the stressed model when that is viewed through the Polariscope. Photoelasticity gives full field map of the principal stress difference. Figure 3.2 shows the fringe pattern obtained by the photoelasticity method for a ring loaded in diametric compression. Moire Interferometry Another optical method that is used is Moire interferometry [2] which is used to measure displacements of a structure undergoing deformation. When two similar but non-identical groups of equally spaced lines (gratings) imposed upon each other, then these lines cause fringe patterns. This effect is called Moire effect. Mechanical or optical means are used for the superposition of two gratings which are either model or specimen gratings and master or reference grating. In Moire method, the model and master gratings are the same and aligned with each other. When the model is subjected to a deformation, the model grating will create fringes. These fringe patterns give a measure of displacement field and hence strain. Shadow Moiré method is used to measure out of plane displacement. Figure 3.3 shows fringe patterns obtained by Moire interferometry method. Others Some other new techniques are Digital image correlation [2], Speckle pattern interferometry [2] and grated fiber obtics method. All these methods are techniques used for experimental analysis of stress. Another powerful method is computational method for obtaining stress pattern using finite element method. These computational techniques used in structural mechanics are an important complement to the experimental stress analysis. Pertinece to Aircraft Design 6 ]] ]] ]] An aircraft undergoes loading from all directions when in flight. Even when the aircraft is on the ground it undergoes loading due to weight of different components of the aircraft. Landing is another time, when a lot of load is applied on the landing gears. These loads causes different types of stresses (as described in previous sections) at different parts of the aircraft structure, which in turns cause strain. Both experimental and computational techniques (described in the previous section) are employed for strain measurements. Experimental study involves, measurement of strain (using one of the techniques above) in the wind tunnel test session and using the data for stress analysis. Also strain measuring devices are deployed for in-flight monitoring for continuous assessment of the aircraft health and for collecting data for future design purpose. Computational techniques are employed for studying stress-strain at those parts of the aircraft, where experimental measurements are very critical or almost impossible and for studying internal stress distributions. So, the question is: why put so much of effort for strain measurements? The answer is pretty obvious: strain beyond the permissible limit of the material causes failure of the aircraft, which may lead to huge loss of lives and other property damages. Figure 4.1 to 4.3 shows some images of failures occuring at the aircraft. So a huge amount of effort is invested in the structural design and analysis of aircraft at the preliminary design step of aircraft design [5]. References # Beer,F.P and Johnston E. R., Mechanics of Materials, 1992. # Dally, W.J and Riley, W.F., Experimental Stress Analysis, 2005. # Raymer, D.P., Aircraft Design: A Conceptual Approach, 2006. # Stern, F.B, Experimental Mechanics, p.221-224, 2006. # Mavris, D, Lecture Slides of AE6343, 2009. # http://images.google.com Solid Mechanics | Continuum Mechanics Category:Solid Mechanics Category:Continuum Mechanics